Step by step
The question
Question: prove by induction that 4ⁿ + 5 is divisible by 3, for every whole number n.
Like dominoes: show it for n = 1, then show that k passes it on to k + 1.
1 Check n = 1
For n = 1: 9, and that is 3 × 3.
Divisible by 3: the first domino falls.
2 Assume it works for k
Suppose it works for k: 4ᵏ + 5 is a multiple of 3.
So it is 3 times some whole number.
Call it m.
Move the rest to the other side: 4ᵏ equals 3m − 5.
We will need this.
3 What we must show
We must show the same for k + 1: every n becomes k + 1.
Careful!
One more power means one more times the number.
It is 4 times as big, not 1 more.
4 Use the assumption
One more power: 4 times the power with k.
Now swap in 3m − 5 for 4ᵏ.
Open the bracket.
Like terms together.
5 Take out 3
Every term has a 3 in it.
Take out 3: 3(4m − 5).
So it works for k + 1 too.
True for n = 1, and passed on each time: true for every whole number n.
The proof is done.
6 Check
Let's check n = 3: it gives 69.
69 is 3 times 23.
Divisible by 3.
Right!
Let's look back.
First: check n = 1.
Then: assume it works for k.
Then: what we must show.
Then: use the assumption.
Then: take out 3.
Last: check.